Does Brownian motion visit every point uncountably many times?
Actually there is an almost sure lower bound on the Hausdorff dimension of the level sets of Brownian motion that follows from the Ray-Knight theorem. $$ a.s \quad \forall a\in \mathbb{R}, \quad \text{dim}\{t \geq 0 | B(t) =a\} \geq \frac{1}{2}. $$
This is a theorem (6.48, p. 170) in the book, "Brownian Motion" by Peres and Morters.